Let $\mathbf { v } ( t ) = t ^ { 2 } \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$ and $\mathbf { w } ( t ) = 2 t \mathbf { i } + 3 \mathbf { j } - t \mathbf { k }$ Then $\frac { d } { d t } ( \mathbf { v } ( t ) \times \mathbf { w } ( t ) )$ is

A) $- 8 \mathbf { i } - \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$
B) $8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } - ( 6 t + 2 ) \mathbf { k }$
C) $8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$
D) $- 8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$
E) $- 8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } - ( 6 t + 2 ) \mathbf { k }$

Question

Let

\mathbf { v } ( t ) = t ^ { 2 } \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }

and

\mathbf { w } ( t ) = 2 t \mathbf { i } + 3 \mathbf { j } - t \mathbf { k }

Then

\frac { d } { d t } ( \mathbf { v } ( t ) \times \mathbf { w } ( t ) )

is

A) $- 8 \mathbf { i } - \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$
B) $8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } - ( 6 t + 2 ) \mathbf { k }$
C) $8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$
D) $- 8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$
E) $- 8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } - ( 6 t + 2 ) \mathbf { k }$

Quizplus · Accepted Answer

The derivative of the cross product of two vectors can be found using the product rule for derivatives, which states $\frac{d}{dt}(\mathbf{u} \times \mathbf{v}) = \frac{d\mathbf{u}}{dt} \times \mathbf{v} + \mathbf{u} \times \frac{d\mathbf{v}}{dt}$. Applying this rule to $\mathbf{v}(t) \times \mathbf{w}(t)$ and calculating the cross products and their derivatives yields the vector $- 8 \mathbf { i } + \left( 3 t ^ { 2 } + 12 t \right) \mathbf { j } + ( 6 t + 2 ) \mathbf { k }$.

Let v(t)=t2i−j+3tk\mathbf { v } ( t ) = t ^ { 2 } \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }v(t)=t2i−j+3tk

Let $\mathbf { v } ( t ) = t ^ { 2 } \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$